Unmanned aerial vehicles (UAVs) have been attracting significant attention because there is a high probability of line-of-sight links being obtained between them and terrestrial nodes in high-rise urban areas. In this work, we investigate cognitive radio networks (CRNs) by jointly designing three-dimensional (3D) trajectory, the transmit power of the UAV, and user scheduling. Considering the UAV's onboard energy consumption, an optimization problem is formulated in which the average achievable rate of the considered system is maximized by jointly optimizing the UAV's 3D trajectory, transmission power, and user scheduling. Due to the non-convex optimization problem, a lower bound on the average achievable rate is utilized to reduce the complexity of the solution. Subsequently, the original optimization problem is decoupled into four subproblems by using block coordinate descent, and each subproblem is transformed into manageable convex optimization problems by introducing slack variables and successive convex approximation. Numerical results validate the effectiveness of our proposed algorithm and demonstrate that the 3D trajectories of UAVs can enhance the average achievable rate of aerial CRNs.